CNR-IEIIT

Information-Based Complexity for Systems and Control: The Probabilistic Setting

Consiglio Nazionale delle Ricerche

Hara Fest at The University of Tokyo @RT 2012

Roberto Tempo

CNR-IEIITConsiglio Nazionale delle Ricerche

Politecnico di Torinotempo@polito.it

CNR-IEIIT Happy Birthday to Shinji!

Hara Fest at The University of Tokyo @RT 2012

CNR-IEIIT Acknowledgments

Research on the probabilistic setting of Information-Based Complexity is joint work with

Fabrizio Dabbene

Hara Fest at The University of Tokyo @RT 2012

Mario Sznaier

CNR-IEIIT

This talk deals with problems which are solvableonly approximately because information is partial

or contaminated

This Lecture - 1

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o co ed

J.F. Traub, G.W. Wasilkowski, H. Wozniakowski, “Information-Based Complexity (IBC)”, 1988

CNR-IEIIT

This talk deals with problems which are solvableonly approximately because information is partial

or contaminated

This Lecture - 2

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o co ed

Objectives:o derive optimal algorithmso compute approximation error o analyze computational complexity

CNR-IEIIT

This talk deals with problems which are solvableonly approximately because information is partial

or contaminated

This Lecture - 3

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o co ed

Different settings:o worst-case (classical)o probabilistic (new)

CNR-IEIIT This Lecture - 4

Different areas may be covered, e.g. information theory,complexity, signal processing, numerical analysis, …

IBC applications: integration problems, solutions ofnonlinear equations, etc

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We are interested in systems and control and in thederivation of optimal algorithms for specific applications

CNR-IEIIT Spaces and Operators

problem element (unknown)Xx

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CNR-IEIIT Spaces and Operators

Xx

exampleconsider input-output pair

problem element (unknown)

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consider input-output pair ξ(x, t) of a dynamic system

with given basis φi(t)

T1

ξ( , ) φ ( ) ( )ni ii

x t x t t x

CNR-IEIIT Spaces and Operators

Xx

problem element (unknown)

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Y

data

I

I x

CNR-IEIIT Spaces and Operators

Xx

problem element (unknown)

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Y

I

I xy = I x + q

data

CNR-IEIIT Spaces and Operators

Xx

information operatorand measurements

problem element (unknown)

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Y

I

I xy = I x + q

and measurementsm n noisy measurements of ξ(x, t) are available for

t1 < t2 < · · · < tmT

1[ ( ) ( )]my t t x q

data

CNR-IEIIT Spaces and Operators

Xx ZS

solution spaceproblem element (unknown)

z = S x

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Y

I

I xy = I x + q

data

CNR-IEIIT Spaces and Operators

Xx ZS

solution spaceproblem element (unknown)

z = S x

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Y

I

I xy = I x + q

solution operatorestimate future values of ξ(x, t) for tm+1 < · · · < tm+s

T1[ ( ) ( )]m m sx t t x S

data

CNR-IEIIT Spaces and Operators

Xx ZS

solution space

zz = S x

problem element (unknown)

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Y

I

I xy = I x + q

A algorithmprovides an estimate

of z = Sxzdata

CNR-IEIIT Ingredients

Problem element x X = Rn with prior information K

Information operator (linear) I: X → Y = Rm (m n)

Information I x corrupted by noise q

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Data y = I x + q

Bounding set Q for q

Solution operator (linear) S: X → Z = Rs (n s)

Algorithm (nonlinear) A: Y → Z

CNR-IEIIT Problem Element x

Problem element (unknown) x K X

K represents prior information (if available)

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CNR-IEIIT Problem Element x

Problem element (unknown) x K X

K represents prior information (if available)

example

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consider input-output pair (x, t) of a dynamic system

T1

ξ( , ) φ ( ) ( )ni ii

x t x t t x

: 0, 1, 2, ,iK x X x i n

CNR-IEIIT Noise q and Bounding Set Q

We assume that

q Q = q: ||q|| Rm

where ||.|| is the lp norm

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CNR-IEIIT Approximation Error

We study the approximation error

where ||.|| is the lp norm

|| ( ) ||x yS A

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w e e ||.|| s e p o

CNR-IEIIT Consistency Set

Consistency set is defined as

1( ) : there exists : = +y x K q Q y x q I I

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CNR-IEIIT Consistency Set I-1(y)

X I-1(y)

ZSS I-1(y)

z

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Y

I

yQ A

CNR-IEIIT

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Worst-Case Setting

CNR-IEIIT Objective for Worst-Case Setting

Objective: construct an algorithm A (nonlinear)= A(y) of z = S x

and compute the worst-case radius rwc(A, y)z

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o prior information x K X (if available)

o data y = I x + q Yo bounding set Q = q: ||q|| Rm

CNR-IEIIT

Worst-Case Radius and Optimal Algorithm

Given data y and the algorithm A, the worst-case radiusis defined as

1

wc

( )( , ) max || ( ) ||

x yr y x y

IA S A

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Given y a worst-case optimal algorithm is defined as

is the worst-case radius of the optimal algorithm

( )x yI

1

wc wc wco o

( )( ) ( , ) inf max || ( ) ||

x yr y r y x y

A IA S A

wcoA

wco ( )r y

CNR-IEIIT Error of the Algorithm

X I-1(y)

ZS

z

z

S I-1(y)

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Y

I

yQ A

z

CNR-IEIIT Worst-Case Optimal Algorithm

X I-1(y)

ZS zS I-1(y)

z

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Y

I

yQ A

CNR-IEIIT

Optimal Algorithms and Chebychev Center

X I-1(y)

ZS ˆcz

|||| overbounding of S I-1(y)

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Y

I

yQis the Chebychev

center of the set S I-1(y)is worst-case optimal

ˆcz

ˆcz

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Probabilistic Setting

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In the probabilistic setting we reduce theconservatism of the worst-case setting

at the expense of a “small” risk

Probabilistic Setting: Conservatism Reduction

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violation functionvo(r) shows how theprobabilistic riskε (0,1) changesas a function ofthe radius r

worst-caseradiusprobabilistic

radiusrisk

CNR-IEIIT Random Noise

Assume that noise q is a random vector with uniform pdfU [Q] and support set Q

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CNR-IEIIT Uniform Density U [Q]

Univariate uniform density

b

1/(b-a)[ , ]a bU

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Multivariate uniform density U [Q]

1 if

vol( )0 otherwise

q QQQ

U

a b

CNR-IEIIT Objective for Probabilistic Setting

Objective: construct an algorithm A (nonlinear)= A(y) of z = S x

and compute the probabilistic radius rpr(A, y, ε)z

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o probabilistic risk ε (0,1)o prior information x K X (if available)

o data y = I x + q Yo random vector q with uniform pdfo bounding set Q = q: ||q|| Rm

CNR-IEIIT Probabilistic Radius

Given data y, accuracy ε (0,1) and the algorithm A, theprobabilistic radius is defined as

1

pr

: ( ) ε ( )\( , ,ε) inf max || ( ) ||

x yr y x y

IA S A

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Remark: we replace the consistency set with

We discard sets Ω of measure μ(Ω) < ε Probabilistic radius is smaller than worst-case radius

( )y

1( )yI

1( ) \y I

CNR-IEIIT Consistency Set I-1(y)

X I-1(y)

ZS

S I-1(y)

z

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Y

I

yQ A

CNR-IEIIT Set

X I-1(y)\Ω

ZS z

1( ) \y I

S I-1(y)

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Y

I

yQ A we discard sets Ωof measure μ(Ω) < ε from consistency set

CNR-IEIIT Probabilistic Optimal Algorithm

Given data y and risk ε (0,1) a probabilistic optimalalgorithm is defined as

1

pr pr pro o : ( ) ε ( )\

( ,ε) ( , ,ε) inf inf max || ( ) ||x y

r y r y x y

A I

A S A

proA

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is the probabilistic radius of optimal algorithm

( ) ( )\x y I

pro ( ,ε)r y

CNR-IEIIT Measure of Sets

Theorem: Let q ∼ U [Q] and K = Rn. Then, for any y ∈ Y

o is uniform 1( )x y I

1 1( ) and ( )y y I S I

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o is log-concave

This result also holds when Q is a convex set

def log-concave:

1( )z x y S S I

1(1 )A B A B

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Violation Function and its Computation

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Violation Function and its Computation

CNR-IEIIT Optimal Violation Function

Given r > 0 and A, define a violation function

1( , ) ( ) : || ( ) ||v r x y x y r A I S A

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Given r > 0, the optimal violation function vo(r) is

1o ( ) inf ( , ) inf ( ) : || ( ) ||v r v r x y x y r

A AA I S A

CNR-IEIIT Chance-Constrained Optimization

Computation of the probabilistic radius of the optimal algorithm

b f l d h i d bl

pr pr pro o( ,ε) ( , ,ε)r y r y A

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may be reformulated as a chance-constrained problem

For any ε (0,1), we can compute solvinga one-dimensional optimization problem in r > 0

pr pro o( , ,ε) min : ( ) εr y r v r A

pr pro( , ,ε)r yA

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Properties of Optimal Violation Function

Theorem: Let q ∼ U [Q], K = Rn and S = identity. Forfixed r > 0, optimal violation is given by

1

o 1

vol ( ) \ :|| ||( ) inf

l ( )cy x x x r

v r

I

I

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This optimization problem is quasi-convex for all well-defined xc (i.e. non-zero volume)

Optimal violation vo(r) is right-continuous and non-increasing for r > 0

o 1( )

vol ( )cx yI

CNR-IEIIT Optimal Violation

I-1(y)X :|| ||x x x r

proof based on the Brunn-Minkovski inequalityfor intersection of convex sets

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X xc :|| ||cx x x r

1

o 1

vol ( ) \ :|| ||( ) inf

vol ( )c

c

x

y x x x rv r

y

I

I

CNR-IEIIT Optimal Violation

I-1(y)

X

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X xc :|| ||cx x x r

1

o 1

vol ( ) \ :|| ||( ) inf

vol ( )c

c

x

y x x x rv r

y

I

I

CNR-IEIIT Extensions and Algorithms

Extensions for non-identity solution operator S

Computation of optimal violation vo(r) requirescomputation of

V l i f l i NP h d 1vol ( ) \ :|| ||cy x x x r I

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Volume computation of polytopes is NP-hard

Two relaxation algorithms:

o Hard - deterministic (SDP-based)

o Soft - probabilistic (randomized-based)

CNR-IEIIT Conclusions

This probabilistic setting is completely new withinsystems and control

Applications:

l i l id ifi i i i

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o classical: identification, estimation, …

o modern: PageRank computation in Google

CNR-IEIIT

(modern): PageRank computation in Google

H. Ishii and R. Tempo“Distributed Randomized

example

Conclusions

* *(1 )( ) mx m H U x

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Algorithms for thePageRank Computation,”IEEE TAC, 2010

x* is PageRank, m=0.15, nis the number of pages, H is

the hyperlink matrix, ∆ represents link failures and

U is a rank-one matrix

(1 )( )x m H U xn

CNR-IEIIT Main References

F. Dabbene and R. Tempo, “Probabilistic and Randomized Toolsfor Control Design,” The Control Handbook (W. S. Levine Ed.),Taylor & Francis, 2010

G. Calafiore, F. Dabbene and R. Tempo “Research onProbabilistic Design Methods,” Automatica, 2011

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g , , R. Tempo, G. Calafiore and F. Dabbene, “Randomized

Algorithms for Analysis and Control of Uncertain Systems,”Springer-Verlag, London, 2005 (second edition in preparation)

http://staff.polito.it/roberto.tempo/